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  <div class="section" id="numpy-convolve">
<h1>numpy.convolve<a class="headerlink" href="#numpy-convolve" title="Permalink to this headline">¶</a></h1>
<dl class="function">
<dt id="numpy.convolve">
<code class="sig-prename descclassname">numpy.</code><code class="sig-name descname">convolve</code><span class="sig-paren">(</span><em class="sig-param">a</em>, <em class="sig-param">v</em>, <em class="sig-param">mode='full'</em><span class="sig-paren">)</span><a class="reference external" href="https://github.com/numpy/numpy/blob/v1.18.1/numpy/core/numeric.py#L720-L817"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#numpy.convolve" title="Permalink to this definition">¶</a></dt>
<dd><p>Returns the discrete, linear convolution of two one-dimensional sequences.</p>
<p>The convolution operator is often seen in signal processing, where it
models the effect of a linear time-invariant system on a signal <a class="reference internal" href="#r95849f33d2b1-1" id="id1">[1]</a>.  In
probability theory, the sum of two independent random variables is
distributed according to the convolution of their individual
distributions.</p>
<p>If <em class="xref py py-obj">v</em> is longer than <em class="xref py py-obj">a</em>, the arrays are swapped before computation.</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><dl class="simple">
<dt><strong>a</strong><span class="classifier">(N,) array_like</span></dt><dd><p>First one-dimensional input array.</p>
</dd>
<dt><strong>v</strong><span class="classifier">(M,) array_like</span></dt><dd><p>Second one-dimensional input array.</p>
</dd>
<dt><strong>mode</strong><span class="classifier">{‘full’, ‘valid’, ‘same’}, optional</span></dt><dd><dl class="simple">
<dt>‘full’:</dt><dd><p>By default, mode is ‘full’.  This returns the convolution
at each point of overlap, with an output shape of (N+M-1,). At
the end-points of the convolution, the signals do not overlap
completely, and boundary effects may be seen.</p>
</dd>
<dt>‘same’:</dt><dd><p>Mode ‘same’ returns output of length <code class="docutils literal notranslate"><span class="pre">max(M,</span> <span class="pre">N)</span></code>.  Boundary
effects are still visible.</p>
</dd>
<dt>‘valid’:</dt><dd><p>Mode ‘valid’ returns output of length
<code class="docutils literal notranslate"><span class="pre">max(M,</span> <span class="pre">N)</span> <span class="pre">-</span> <span class="pre">min(M,</span> <span class="pre">N)</span> <span class="pre">+</span> <span class="pre">1</span></code>.  The convolution product is only given
for points where the signals overlap completely.  Values outside
the signal boundary have no effect.</p>
</dd>
</dl>
</dd>
</dl>
</dd>
<dt class="field-even">Returns</dt>
<dd class="field-even"><dl class="simple">
<dt><strong>out</strong><span class="classifier">ndarray</span></dt><dd><p>Discrete, linear convolution of <em class="xref py py-obj">a</em> and <em class="xref py py-obj">v</em>.</p>
</dd>
</dl>
</dd>
</dl>
<div class="admonition seealso">
<p class="admonition-title">See also</p>
<dl class="simple">
<dt><a class="reference external" href="https://docs.scipy.org/doc/scipy/reference/generated/scipy.signal.fftconvolve.html#scipy.signal.fftconvolve" title="(in SciPy v1.4.1)"><code class="xref py py-obj docutils literal notranslate"><span class="pre">scipy.signal.fftconvolve</span></code></a></dt><dd><p>Convolve two arrays using the Fast Fourier Transform.</p>
</dd>
<dt><a class="reference external" href="https://docs.scipy.org/doc/scipy/reference/generated/scipy.linalg.toeplitz.html#scipy.linalg.toeplitz" title="(in SciPy v1.4.1)"><code class="xref py py-obj docutils literal notranslate"><span class="pre">scipy.linalg.toeplitz</span></code></a></dt><dd><p>Used to construct the convolution operator.</p>
</dd>
<dt><a class="reference internal" href="numpy.polymul.html#numpy.polymul" title="numpy.polymul"><code class="xref py py-obj docutils literal notranslate"><span class="pre">polymul</span></code></a></dt><dd><p>Polynomial multiplication. Same output as convolve, but also accepts poly1d objects as input.</p>
</dd>
</dl>
</div>
<p class="rubric">Notes</p>
<p>The discrete convolution operation is defined as</p>
<div class="math">
<p><img src="../../_images/math/ad06f7bde7ec2032ae21a0accc68d696eef7bd8e.svg" alt="(a * v)[n] = \sum_{m = -\infty}^{\infty} a[m] v[n - m]"/></p>
</div><p>It can be shown that a convolution <img class="math" src="../../_images/math/abb88b244ac16c2ed2c58f73c8a639a5e49916f5.svg" alt="x(t) * y(t)"/> in time/space
is equivalent to the multiplication <img class="math" src="../../_images/math/ab559a702530db5f848b610d5b35a57c6d0afc28.svg" alt="X(f) Y(f)"/> in the Fourier
domain, after appropriate padding (padding is necessary to prevent
circular convolution).  Since multiplication is more efficient (faster)
than convolution, the function <a class="reference external" href="https://docs.scipy.org/doc/scipy/reference/generated/scipy.signal.fftconvolve.html#scipy.signal.fftconvolve" title="(in SciPy v1.4.1)"><code class="xref py py-obj docutils literal notranslate"><span class="pre">scipy.signal.fftconvolve</span></code></a> exploits the
FFT to calculate the convolution of large data-sets.</p>
<p class="rubric">References</p>
<dl class="citation">
<dt class="label" id="r95849f33d2b1-1"><span class="brackets"><a class="fn-backref" href="#id1">1</a></span></dt>
<dd><p>Wikipedia, “Convolution”,
<a class="reference external" href="https://en.wikipedia.org/wiki/Convolution">https://en.wikipedia.org/wiki/Convolution</a></p>
</dd>
</dl>
<p class="rubric">Examples</p>
<p>Note how the convolution operator flips the second array
before “sliding” the two across one another:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">np</span><span class="o">.</span><span class="n">convolve</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">])</span>
<span class="go">array([0. , 1. , 2.5, 4. , 1.5])</span>
</pre></div>
</div>
<p>Only return the middle values of the convolution.
Contains boundary effects, where zeros are taken
into account:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">np</span><span class="o">.</span><span class="n">convolve</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">],[</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mf">0.5</span><span class="p">],</span> <span class="s1">&#39;same&#39;</span><span class="p">)</span>
<span class="go">array([1. ,  2.5,  4. ])</span>
</pre></div>
</div>
<p>The two arrays are of the same length, so there
is only one position where they completely overlap:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">np</span><span class="o">.</span><span class="n">convolve</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">],[</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mf">0.5</span><span class="p">],</span> <span class="s1">&#39;valid&#39;</span><span class="p">)</span>
<span class="go">array([2.5])</span>
</pre></div>
</div>
</dd></dl>

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